Evaluation of E-Learning Web Sites Using Fuzzy Axiomatic Design Based Approach

1
Büyüközkan, G. and Arsenyan, J. and Ertek, G. (2010).“Evaluation of e-learning web sites
using fuzzy axiomatic design based approach.” International Journal of Computational
Intelligence Systems, 3 (1). pp. 28-42.
Note: This is the final draft version of this paper. Please cite this paper (or this final draft)
as above. You can download this final draft from http://research.sabanciuniv.edu.
Evaluation of E-Learning Web Sites
Using Fuzzy Axiomatic Design Based Approach
Gülçin Büyüközkan
Department of Industrial Engineering, Galatasaray University
Çırağan Caddesi No: 36 Ortaköy, İstanbul, 34357, Turkey
www.gsu.edu.tr
Jbid Arsenyan
Department of Industrial Engineering, Bahçeşehir University
Çırağan Caddesi Osmanpaşa Mektebi Sokak No: 4 – 6, Beşiktaş, İstanbul, 34100, Turkey
www.bahcesehir.edu.tr
Gürdal Ertek
Faculty of Engineering and Natural Sciences, Sabancı University
Orhanlı, Tuzla, İstanbul, 34956, Turkey
www.sabanciuniv.edu

Abstract
High quality web site has been generally recognized as a critical enabler
to conduct online business. Numerous studies exist in the literature to
measure the business performance in relation to web site quality. In this
paper, an axiomatic design based approach for fuzzy group decision
making is adopted to evaluate the quality of e-learning web sites. Another
multi-criteria decision making technique, namely fuzzy TOPSIS, is
applied in order to validate the outcome. The methodology proposed in
this paper has the advantage of incorporating requirements and enabling
reductions in the problem size, as compared to fuzzy TOPSIS. A case
study focusing on Turkish e-learning websites is presented, and based on
the empirical findings, managerial implications and recommendations
for future research are offered.
Keywords: Fuzzy axiomatic design, Group decision making, Web site
quality, E-Learning web sites, Fuzzy TOPSIS.
1. Introduction
E-Learning, one of the e-service applications, is a wide set of applications and processes
that manage diverse types of electronic media to deliver vocational education and training
(Aladwani and Palvia, 2002). For e-learning service providers, the Internet serves as the
primary interface with the e-learners, since the e-learning web site has a much more
extended function, compared to conventional web sites, which only disseminate
information about services and products. Consequently, the web site quality should be
considered as a critical success factor for e-learning service providers. Especially, in the
case of vital education or training services, the web site quality and its evaluation should be
studied in a more detailed manner from e-learners’ perspective (Colette, 2001).
In a number of publications, quantitative methods are adopted for the evaluation of web
site quality, with statistical methods ranking as the most widely used assessment tools

(Chao, 2002; Cox and Dale, 2002; Jeong et al., 2003; Kim et al., 2003; Kim and Stoel,
2004; Toms and Taves, 2004). Additionally, other methods such as multidimensional
scaling and correspondence analysis (Van der Merwe and Bekker, 2003), weighted scores
(Barnes and Vidgen, 2003), index method (González and Palacios, 2004), soft computing
technologies (Hwang et al., 2004) and multi criteria decision making (MCDM) (Bilsel et al.,
2006) are also used in assessing and improving the web site quality. Nonetheless, there
exist few studies comparing customer needs to web sites performance. Axiomatic Design
(AD) principles (Suh, 2001) provide a powerful tool to measure how well system
capabilities respond to functional requirements. The ultimate goal of AD is to establish a
scientific basis for design and to improve design activities. This is achieved through
providing the designer with a theoretical foundation based on logical and rational thought
process and tools. AD applications include a multitude of areas such as software design
(Kim et al., 1991), quality system design (Suh, 1995a), general system design (Suh, 1995b;
Suh, 1997), manufacturing system design (Suh et al., 1998; Cochran et al., 2001),
ergonomics (Helander and Lin 2002), engineering systems (Guenov and Barker, 2005;
Thielman and Ge, 2006), office cell design (Durmusoglu and Kulak, 2008). Even though
AD is traditionally applied to the design of physical entities, there exist studies that employ
AD in designing intangible systems, such as e-commerce strategies (Martin and Kar, 2002)
and e-commercial web sites (Yenisey, 2007).
Conventional information content approach cannot be used in the case of incomplete
information, since, the expression of system and design ranges by crisp numbers would be
ill defined (Kahraman and Kulak, 2005). For this reason, under incomplete information,
the subjectivity and vagueness in the assessment process is dealt with fuzzy logic (Zadeh,
1975). The information axiom of AD is utilized as a fuzzy MCDM technique by Kulak and
Kahraman (2005a). However, while there exist many applications of AD methodology (Suh,
2001) in literature, there are relatively few studies on fuzzy AD applications for MCDM.
Studies in this domain can be summarized as follows:
In two pioneering studies, Kahraman and Kulak (2005a, 2005b) apply fuzzy AD
approach to the comparison of advanced manufacturing systems and then to the multi-
attribute selection among transportation companies. Kulak (2005) develops a decision

support system for the selection of material handling systems, based on fuzzy AD.
Kahraman and Cebi (2009) propose a hierarchical fuzzy AD model, which they apply to
teaching assistant selection problem. Celik et al. (2009a) employ the method for shipyard
selection. They also utilize fuzzy AD and Fuzzy TOPSIS to manage strategies on Turkish
container ports in maritime transportation and then apply SWOT (Strengths, Weaknesses,
Opportunities, Threats) analysis to the outcome of the two techniques (Celik et al., 2009b).
In another study, the authors integrate fuzzy AD and fuzzy AHP into QFD (Quality Function
Deployment) principles for routing of shipping investment decisions in crude oil tanker
market (Celik et al., 2009c). Celik (2009) applies fuzzy AD methodology along with AHP in
order to combine management standards for ship management companies and Celik et al.
(2009) employ the method for shipyard selection. Recently, Yücel and Aktas (2008)
propose an evaluation methodology for ergonomic design of electronic consumer products
based on fuzzy AD approach while Cevikcan et al. (2009) utilize fuzzy AD technique for an
application of candidate assessment.
The aim of this paper is to attain a group consensus on functional requirements of an
ideal e-learning web site. A case study is then conducted in order to evaluate several e-
learning web sites according to these functional requirements with group fuzzy AD. Fuzzy
AD methodology is based on the conventional AD; however, crisp ranges are replaced by
fuzzy numbers that represent linguistic terms. For measuring intangible criteria such as
reliability, responsiveness, etc., fuzzy AD is applied to translate linguistic terms into
performance measures. Also, group consensus is sought throughout the study and
therefore, fuzzy AD model is enhanced with a group decision making tool.
The paper is organized as follows. In next section, e-learning web site evaluation criteria
are defined. Section 3 briefly describes the proposed fuzzy AD based evaluation
methodology. A case study is conducted in e-learning web sites evaluation and the
outcomes are explained in Section 4. The concluding remarks are given in the last section.
2. Evaluation criteria for e-learning web sites
Internet-oriented applications aim at satisfying current educational needs by closing the
gap between traditional educational techniques and future trends in technology-blended

education (Tzouveli et al., 2008), enabling a new type of education on online platforms. E-
Learning refers to Internet technologies used to deliver a broad array of solutions that
support the instructional process in a networked environment through the establishment of
an interactive virtual classroom (Poon et al., 2004). The expected outcomes of online
teaching and learning are largely dependent on the quality of the teaching processes and the
effectiveness of online access. To this end, e-learning systems must be designed and
constructed cautiously, especially while applying a scientific approach with well-designed
procedures and techniques. The ultimate goal is to accomplish an effective and high quality
learning system, comparable with the traditional educational systems (Colette, 2001). Web
sites appear as the primary interface to the end user (e-learner) and user satisfaction vis-à-
vis human-computer interaction determines the quality of the e-learning provider. An
organization with a poor web site or ineffective services may project weaken the
organization’s image and position. Hence, determining evaluation criteria for e-learning
web sites is important in order to determine user needs (Ahn et al., 2007). In this context,
an e-learning web site quality has to be analyzed in a more detailed manner.
Literature offers numerous studies investigating e-service and e-learning web site
evaluation criteria. Webb and Webb (2004) states that a business to customer (e-learning
provider to e-learner, in our context) web site quality is directly affected by service quality
and information quality. According to Ahn et al. (2007), even though web site evaluation
criteria may vary, the main categories include system, information, and service quality.
System quality (such as interface design and functionality), is an engineering oriented
performance characteristic while information quality (such as completeness and timeliness)
has both engineering and operational characteristicsService quality refers to availability of
communication, mechanisms for accepting consumer complaints and their timely
resolution with responsiveness, assurance, and follow-up services. According to the survey
conducted by Poon et al. (2004), five main factors influence the effectiveness of e-learning
process: students’ behavior, characteristics of lecturers, interactive application, technology
or system, and the institutions. On the other hand, Mahdavi et al. (2008) state that e-
learner satisfaction can be classified into four dimensions: content, personalization,
learning community, and learner interface. Kim and Lee (2008) detect two principle factors

for learning management systems. Factor I consists of instruction management, screen
design, and technology; whereas Factor II consists of interaction and evolution. McPherson
and Nunest (2008) investigate the critical success factors required to deliver e-learning
within higher education programs and they cite five fundamental aspects of e-learning:
organizational, technological, curriculum design, instructional design and e-learning course
delivery.
Based on an in-depth literature analysis [such as (Smith, 2001; Aladwani and Palvia,
2002; Chao, 2002; Cox and Dale, 2002; Dragulanescu, 2002; Jeong et al., 2003; Kim et al.,
2003; van der Merwe and Bekker, 2003; Wang, 2003; Hwang et al., 2004; Kim and Stoel,
2004; van Iwaarden et al., 2004; Webb and Webb, 2004; Barnes and Vidgen, 2006;
Büyüközkan et al., 2007; Gonzalez et al., 2007; Grigoroudis et al., 2008; van den Haak et
al., 2009)], results of industrial surveys and in the light of the expert suggestions, seven
main criteria were determined as the e-learning web site quality dimensions in this study.
Ahn et al. (2007) state that technology-focused approach considers the web site as an
information system, while service-focused approach sees a web site as a service provider.
Following criteria were determined with a point of view combining the two approaches:
 Right and Understandable Content (C1): This criterion includes credibility,
clearness and succinctness. While using educational web sites, authority is a particular
concern, as high quality content must be assured. Instructional objectives should also be
assured. In addition, the content should be easily understood, unambiguous and
succinct.
 Complete Content (C2): This criterion includes accuracy and coverage. The purpose
of this assessment is to guarantee that the content is actually correct: up to date, factual,
detailed, exact and comprehensive. This criterion also assesses the existence of tests,
quizzes and exams for adequate evaluation procedures.
 Personalization (C3): This dimension states a level of individualization. This can
make the web site more attractive for the e-learners.

 Security (C4): This dimension comprises criteria that may be used for evaluating the
security of a web site. A confident web site should assure the secrecy of its users’
personal and private data. The scope of the privacy should be stated in the web site. In
order to place such information in the web site, having a digital certificate is desirable.
 Navigation (C5): This criterion describes the ability of web-based service systems to
perform the online service consistently and accurately. It controls the organization and
technical capabilities of the navigation through the pages.
 Interactivity (C6): This dimension measures the availability of complementary
functions of the traditional communication media to digital media. Availability of
Frequently Asked Questions (FAQs), help and feedback systems constitute the content
of this dimension. Adequate responsiveness is an important source of motivation for the
e-learners.
 User Interface (C7): This criterion includes the design appearance, consistency, the
information structure and the organization of the web site. Applications of the right
design principles are essential. A consistent interface allows the e-learners to follow the
required tasks easily. The information structure and organization of the web site should
also be easy to follow and to be understood by the e-learners.
3. Fuzzy Axiomatic Design based Group Decision-Making
In line with the multi-dimensional characteristics of web site quality, MCDM methodology
is a powerful tool widely used for evaluating and ranking problems containing multiple,
usually conflicting criteria. Over the years, several behavioral scientists, operational
researchers and decision theorists have proposed a variety of methods describing how an
evaluator might arrive at a preference judgment while choosing among the multiple
alternatives. Hence, this work attempts to model the e-learning web site evaluation in an
MCDM framework. In addition, the subjectivity and vagueness in the assessment process is
dealt with fuzzy logic (Zadeh, 1975). Multiple decision makers (DMs) are often preferred
rather than a single DM to avoid the bias and to minimize the partiality in the decision
process (Herrera et al., 2001). Therefore, fuzzy MCDM with group decision is increasingly
employed in literature, as evaluation criteria become more intangible and the decision

making becomes more complex to make for single DM. For example, Chen and Cheng
(2005) apply fuzzy MCDM with group decision to information systems personnel selection.
Wang and Parkan (2008) consider fuzzy preference aggregation problem in group decision
and they apply it to the broadband internet service selection. Recently, Yeh and Chang
(2009) develop a hierarchical weighting method in order to assess the weights of a large
number of evaluation criteria by pairwise comparisons.
This paper proposes a set of evaluation criteria for e-learning web sites, as well as a
methodology to evaluate these web sites. Main steps of the proposed methodology are
recapitulated in Figure 1. The first step in the methodology is determining e-learning web
site evaluation criteria. In this study, criteria described in Section 2 are employed. These
criteria undergo pairwise comparison by a group of DMs. Fuzzy Analytic Hierarchy Process
(AHP) is then applied to compute the criteria weights. E-learning web site alternatives are
identified and several sites are considered in order to cover all available services on the net.
Then, alternatives and functional requirements are evaluated by DMs. These evaluations
are translated into fuzzy numbers and then are aggregated. Information contents are
calculated accordingly and alternatives that cannot meet the functional requirements are
eliminated. The last step of fuzzy AD methodology is ranking the alternatives in respect to
weighted information contents and selecting the best web site according to a decreasing
order of information content. Finally, fuzzy TOPSIS technique is applied in order to
compare the outcome of two methodologies.
Techniques employed in the study, namely Fuzzy AD, fuzzy AHP, Chen’s aggregation
methodology and fuzzy TOPSIS are now described.
3.1.Fuzzy Axiomatic Design
AD, a systematic method offering a scientific base for design, was introduced by Suh (1990)
and its application areas include software design, quality system design, general system
design, manufacturing system design, ergonomics, engineering systems, office cell design,
and e-commerce strategies. AD is based on two axioms. The independence axiom states
that the independence of functional requirements should be maintained and information
axiom states that among the designs that satisfy the functional requirements, the design

with the minimum information content is the best design. Information content, on which
MCDM technique is based, represents a function of probability of satisfying a functional
requirement . Therefore, the design with the highest probability to meet these
requirements is the best design. Information content of a design with probability of
success for a given is defined as follows:

Fig. 1. This is the caption for the figure. If the caption is less than one line then it is
centered. Long captions are justified manually.
Literature
survey
Expert
experiences
Web
search
Step 1. Determining e-learning website evaluation criteria
Step 3. Determining e-learning web site alternatives
Step 4.a. Evaluating of web site alternatives
Step 4.b. Evaluating of functional requirements
Step 5. Fuzzification and aggregation of experts’ opinions
Step 2. Determining criteria weights with fuzzy AHP
Expert
opinions
Expert
opinions
Expert
opinions
Expert
opinions
Step 6. Calculating information contents
Step 7. Calculating weighted information contents
Step 8. Ranking of the e-learning web site alternatives
Step 9. Comparing the obtained results with fuzzy TOPSIS
Expert
opinions
FAD
implementation

= log (1)
According to Suh (2001), logarithm is employed in calculating the information contents,
so as to attain additivity.
On the other hand, the probability of success is given by the design range (the
requirements for the design) and the system range (the system capacity). Figure 2
illustrates the design and system ranges as well as the common area. The intersection of the
ranges offers the feasible solution. Therefore, the probability of success can be expressed as:
= ∫ ( ) (2)
where and represent the lower and upper limits of the design range and where
represents the probability distribution function of the system for a given .
The probability of success is equal to the common area . Consequently, the information
content can be expressed as follows:
= log (3)
Also, if the probability distribution function is uniform, the probability of success becomes:
=

(4)
Therefore, the information content can also be written as:
= log

(5)
Fuzzy AD methodology is based on the conventional AD. However, crisp ranges are
replaced by fuzzy numbers that represent linguistic terms (Figure 3). In this study,
triangular fuzzy numbers (TFNs) are employed. Intersection of TFNs representing design
and system ranges presents the common area (Kulak and Kahraman, 2005a). Firstly, the
information content is calculated as in a non-fuzzy environment. Then information content
in a fuzzy environment is calculated as follows:

=
∞ ,
log

, ℎ
(6)
In this study, the calculation of the weighted information content is adapted from
Kahraman and Cebi (2008). This model requires determination of weights of criteria and
sub-criteria. Total weighted information content for first level criteria is calculated as
follows:
= ∑ (7)
where n is the number of first level criteria and ∑ = 1.
Fig. 3. System-design ranges and common
area in fuzzy environment
Fig. 2. System-Design ranges and
common area.

Likewise, information content for second level criteria (sub-criteria for criterion ) is
calculated as follows:
= ∑ (8)
where is the number of sub-criteria for criterion and ∑ = 1 for = 1, … .
Finally, according to information axiom, alternatives are ranked with increasing order of
information content.
3.2. Fuzzy AHP
It is not possible to assume that e-learning website evaluation criteria are of equal
importance. There are many methods that can be employed to determine weights, such as
eigenvector, weighted least square, entropy methods and diverse MCDM methods. In this
study, the fuzzy extension of the one of the most outstanding MCDM approaches, AHP
(Saaty, 1980) is used to determine the decision criteria weights. Despite its wide range of
applications, the conventional AHP approach may not fully reflect a style of human
thinking. One reason is that decision makers usually feel more confident to give interval
judgments rather than expressing single numeric values. Furthermore, knowledge concepts
and models contain tacit values and uncertainty, causing assessment and evaluation to be
more difficult. This difficulty is handled by applying AHP in fuzzy environment to solve
prioritisation and evaluation problems.
Firstly advocated by Zadeh (1965), fuzzy set theory has become important to deal with
the ambiguity in a system. In this study, firstly linguistic terms are used to represent the
expert assessments, then triangular fuzzy numbers, 1 to 9 as given in Table 1, are used to
represent subjective pair-wise comparisons of evaluation processes in order to capture the
vagueness.

The four step computational procedure is given as follows:
1. Compare the performance score. Triangular fuzzy numbers (1, 3, 5, 7, 9) are used to
indicate the relative strength of each pair of elements in the same hierarchy.
2. Construct the fuzzy comparison matrix. By using triangular fuzzy numbers, via pair-
wise comparison, the fuzzy judgment matrix ( ) is constructed as given below:
=
1 ⋯
1 ⋯
⋮ ⋮ ⋱
⋯
⋮
1
(9)
where ãα
ij = 1, if is equal to , and ãα
ij = 1, 3, 5, 7, 9 or 1 , 3 , 5 , 7 , 9 , if is not equal
to .
3. Solve the fuzzy eigenvalue. A fuzzy eigenvalue, λ̃, is a fuzzy number solution to:
= (10)
where is a × fuzzy matrix containing fuzzy numbers and is a non-zero × 1 fuzzy
vector containing fuzzy number . To perform fuzzy multiplications and additions by using
the interval arithmetic and α-cut, the equation = is equivalent to:
[ , ] ⊕ ⋯ ⨁[ , ]
= [ , ] (11)
Table 1. Definition and membership function of fuzzy number (Saaty, 1989)
Intensity of Importance Fuzzy Number Definition Membership Function
9 9̃ Extremely more importance (EMI) (8,9,10)
7 7̃ Very strong importance (VSI) (6,7,8)
5 5̃ Strong importance (SI) (4,5,6)
3 3̃ Moderate importance (MI) (2,3,4)
1 1̃ Equal importance (EI) (1,1,2)

where:
= , = ( , … , ), = , ,
= [ , ], = [ , ] (12)
for 0 < ≤ 1 and all , , with = 1, 2, … , , = 1, 2, … , .
The α-cut is known to incorporate the experts or decision-maker(s) confidence over
his/her preference or the judgments. The degree of satisfaction for the judgment matrix is
estimated by the index of optimism . A larger value of the index indicates a higher degree
of optimism. The index of optimism is a linear convex combination (Lee, 1999) defined as:
= + (1 − ) , ∀ ∈ [0,1] (13)
When is fixed, the following matrix can be obtained after setting the index of optimism,
, in order to estimate the degree of satisfaction:
=
1 ⋯
1 ⋯
⋮ ⋮ ⋱
⋯
⋮
1
(14)
The eigenvector is calculated by fixing the value and identifying the maximal
eigenvalue. After defuzzification of each pair wise matrix, the consistency ratio ( ) for
each matrix is calculated. The deviations from consistency are expressed by the following
equation consistency index, and the measure of inconsistency is called the consistency
index ( ):
= (15)
The consistency ratio ( ) is used to estimate directly the consistency of pair wise
comparisons. The is computed by dividing the by a value obtained from a table of
Random Consistency Index ( ) to:
= (16)
If is less than 0.10, the comparisons are acceptable, otherwise not. is the average

index for randomly generated weights (Saaty, 1980).
4. The priority weight of each criterion can be obtained by multiplying the matrix of
evaluation ratings by the vector of attribute weights and summing over all attributes.
3.3. Aggregation methodology
In this study, the fuzzy group decision-making method presented by Chen (1998) is
employed in order to aggregate fuzzy opinions of the DMs. This method is recently
employed by Celik et al. (2009).
The steps of the aggregation method are as follows:
1. Calculate the degree of agreement , ( , )of the opinions between each pair of
experts and where , ( , ) ∈ [0,1], 1 ≤ ≤ , 1 ≤ ≤ and ≠ .
Let and be two standardized triangular fuzzy numbers = ( , , ), = ( , , )
where 0 ≤ ≤ ≤ ≤ 1 and 0 ≤ ≤ ≤ ≤ 1.
Then the degree of similarity between the standardized triangular fuzzy numbers and
can be measured by the similarity function:
( , ) = 1 −
| | | | | |
(17)
where ( , ) ∈ [0,1]. The larger the value of ( , ), the greater is similarity between the
standardized triangular fuzzy numbers and . The equation ( , ) = ( , ) is valid for
the degree of similarity.
2. Calculate the average degree of agreement ( ) of expert , = 1,2, … , , where:
( ) = , ( , ),
(18)
3. Calculate the relative degree of agreement ( ) of expert , = 1,2, … , , where:
( ) =
( )
∑ ( )
(19)
4. Calculate the consensus degree coefficient ( ) of expert , = 1,2, … , where:
( ) = + (1 − ) ( ) (20)

(0 ≤ ≤ 1) is a relaxation factor of the method and is degree of importance of
expert. It shows the importance of over ( ).
5. The aggregation result of the fuzzy opinions is
= ( )⨂ ⨁ … ( )⨂ (21)
where operators ⨂ and ⨁ are the fuzzy multiplication operator and the fuzzy addition
operator, respectively. The method is independent of the type of membership functions
being used (Chen, 1998; Celik et al., 2009).
3.4. Fuzzy TOPSIS
In our methodology, another MCDM method, namely TOPSIS is applied in order to
compare with fuzzy AD outcome. TOPSIS is proposed by Chen and Hwang (1992) and the
basic principle is that the optimal solution should have the shortest distance from the
positive ideal solution and the farthest from the negative ideal solution.
In classical MCDM methods, including classical TOPSIS, all data are assumed to be
known precisely. However, under many conditions, crisp data are inadequate to model real-
life situations since human judgments including preferences are often vague and cannot be
estimated with an exact numerical value (Saghafian and Hejazi, 2005). Linguistic terms
present a more realistic assessment of subjective judgments and hence, fuzzy set theory aids
to deal with biased or imprecise evaluations.
There are many examples of applications of fuzzy TOPSIS in literature (Saghafian and
Hejazi, 2005) such as evaluation of service quality (Tsuar et al., 2002), inter company
comparison (Deng et al., 2000), aggregate production planning (Wang et al., 2004), facility
location selection (Chu, 2002) and large scale nonlinear programming (Abo-Sina and
Amer, 2005).
Fuzzy TOPSIS is applied in this study due to its basic concept and wide applications such
as Qureshi et al. (2008) and Shih (2008). Moreover, TOPSIS is based on geometrical
principles, similar to AD which also operates on a geometrical level. The technique is
adapted from Chen (2000) and the steps of the methodology are as follows:

1. With alternatives, criteria and DMs, fuzzy MCDM problem can be expressed
as:
=

⋯
⋮ ⋮
⋮
⋯ ⋯
⋮
represents the fuzzy decision matrix with alternatives and criteria .
2. Aggregated judgments are calculated as follows is
= ( + + ⋯ + ) (22)
where = ( , , ̃ ) represents fuzzy judment of expert .
3. The next step is the normalization. Normalized fuzzy decision matrix is calculated as
= ̃
∗
, = 1,2, …, ; = 1,2, …,
= , , , = max (23)
To avoid the complicated normalization formula used in classical TOPSIS, the linear
scale transformation is used to transform the various criteria scales into a comparable scale
(Chen, 2000). Linear scale transformation for normalization is also employed by Kuo et al.
(2007) and Celik et al. (2009).
Table 5. Membership functions for system and design ranges.
Term Abbr. Membership Term Abbr. Membership
Poor P 0, 0, 0.3 At least poor LP 0, 1, 1
Fair F 0.2, 0.35, 0.5 At least fair LF 0.1, 1 1
Good G 0.4, 0.55, 0.7 At least good LG 0.4, 1, 1
Very good VG 0.6, 0.75, 0.9 At least very good LVG 0.6, 1, 1
Excellent E 0.8, 1, 1 At least excellent LE 0.8, 1, 1

4. Then, weighted normalized fuzzy decision matrix is computed, where is weight
for criteria :
= ⨂
=
∗
, = 1,2, … , ; = 1,2, … , (24)
5. Since the TFNs are included in [0,1] range, positive and negative ideal reference
points (FPIRP, FNIRP) are as follows:
= { , , … , }, = { , , … , }
where = (1,1,1), = (0,0,0).
6. The next step is calculating the distance of alternatives from FPIRP and FNIRP:
, = [( − ) + ( − ) + ( − ) ] (25)
= ∑ ( , ) , = 1,2, … , ; = 1,2, … , (26)
= ∑ ( , ) , = 1,2, … , ; = 1,2, … , (27)
7. The performance indices are computed in order to rank the alternatives. Performance
indices are sorted in decreasing order
= , = 1,2, … , (28)
4. Case study: Evaluation of e-learning web sites

E-Learning became an instructional delivery method for the growing number of working
adults who sought to earn degrees from universities that provide external programs (Poon
et al., 2004). Finance, time or access constraints are minimized with e-learning and an
equal-opportunity education environment is generated, since e-learning offers cost-
effectiveness, timely content and access flexibility to e-learners (Mahdavi et al., 2008).
Turkey, a country of 783,562 km2 facing the challenge of providing the same quality
education nationwide, benefits from this equal-opportunity environment. E-Learning is
classically considered to be a new way to empower the workforce with the necessary skills
and knowledge (Tzouveli et al., 2008); However, considering the special case of Turkey,
where the universities are concentrated on major cities, e-learning stands out as the new
era’s education provider for not only the workforce, but also for the disadvantaged youth.
The current demographics of Turkey where 30.64% of the population (20,778,277 citizens)
is made up of 10 to 24-year-olds (Statistics Institute of the Government of Turkey, 2000)
engender a very high number of candidates for university education. The annual quota is
500,000 whereas the number of university applicants exceeds 1,600,000 (Student
Placement Center, 2008). The severe negative impacts of the capacity constratint, as
reflected by the quotas, can be overcome through e-learning, which proposes a great
potential to face this educational challenge. Today, nearly all Turkish universities have their
own web sites and hence, they seize an incredible opportunity to catch up to developed
countries (Kızılsu, 2006). E-Learning activities are broadening within the universities, as
well. Many e-programs are executed at the graduate level, and young professionals with
time limit and individuals with no access to major cities can profit from the educational
Fig. 4. Membership Functions for System
Range.

added-value of online access.
However, as the number of available online programs increases, the decision process
becomes complex. Therefore, the need arises to evaluate the quality of e-learning providers
as a pre-requisite of achieving high quality of e-learning in Turkey. In this study web sites of
e-learning providers are considered given that the interface greatly influences the e-learner
satisfaction with the e-learning system.
The methodology described in Figure 1 is applied to a case study. The goal of this case
study is to evaluate and rank the quality performances of e-learning web sites, with the
proposed methodology. The web sites are selected from among successful actors operating
globally and locally in Turkey. To identify the functional requirements and evaluate the
alternatives, three DMs of equal importance, DM1, DM2 and DM3, have been selected
amongst e-learning industry experts. These experts are gathered from knowledgeable e-
learning instructors involved in educational design and implementation of online
interfaces. They possess an extended experience in e-learning systems given that they have
been the pioneers of the industry.
The e-learning web site evaluation process is performed by applying the following steps:
Step 1. Determination of e-learning web site evaluation criteria
Right and understandable content (C1), Complete content (C2), Personalization (C3),
Security (C4), Navigation (C5), Interactivity (C6) and User interface (C7) are the e-learning
web site evaluation criteria as discussed in Section 2.
Step 2. Determination of criteria weights for e-learning web site evaluation
DMs apply pairwise comparison to evaluate criteria as given in Table 2 and fuzzy AHP
given in Section 3.2 is employed with the index of optimism value = 0.5 in order to
determine e-learning web site criteria weights. The obtained results are shown in Table 3.
Step 3. Determination of alternatives
Table 2. The consensus linguistic comparison matrix for e-learning web site
evaluation criteria.

C1 C2 C3 C4 C5 C6 C7
C1 1 P
C2 1
C3 F P 1 VG
C4 G F E 1 E
C5 G P E F 1 E G
C6 G F E F 1 G
C7 E G 1
Table 3. E-Learning web site evaluation criteria weights.
C1 C2 C3 C4 C5 C6 C7
0.15 0.33 0.08 0.14 0.07 0.08 0.15
Table 4. E-Learning web site alternatives.
Label Web address
W1 www.online-degree-enlightenment.com
W2 www.youachieve.com
W3 www.online-education-resources.com
W4 www.universalclass.com
W5 www.sp.edu.sg
W6 www.geolearning.com
W7 www.kidsplus.com.tr
W8 www.ideaelearning.com
W9 www.sanal-kampus.com

W10 www.netron.com.tr
W11 www.enocta.com
W12 www.buelc.boun.edu.tr
To assess the quality of e-learning web sites, 12 web sites given in Table 4 were
designated, taking into account experts’ opinions in the sector and research conducted
through the search engines in January 2008. The first six of these web sites operate
worldwide and the remaining six are active only in Turkey. Since the sites all offer mostly
common services, the comparison is coherent.
Step 4. DMs’ evaluation of web site alternatives and functional requirements
Linguistic terms employed in evaluating e-learning web sites needs to be translated into
fuzzy numbers in order to operate on the judgments. In this study, 5-level fuzzy scale is
used to assess the alternatives and another 5-level fuzzy scale is used to assess the
Fig. 5. Membership Functions for Design
Range.
Table 6. Evaluation of alternatives by DM1.
W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12
C1 VG VG VG F VG VG VG VG VG G VG G
C2 E VG E F VG F G VG E G E F
C3 F G G VG G F F G VG G E VG
C4 P VG E G VG G VG F E VG E VG
C5 F VG VG G F F VG VG VG F VG G
C6 F E VG G G G G G F F E G
C7 F VG G G VG VG VG G VG F E G
W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12
C1 G E VG G E E E E VG F E F
C2 VG E VG F E F VG VG VG G VG F
C3 P G VG E G F F VG VG F VG VG
C4 F VG E VG VG G VG F VG G E G
C5 F VG E VG F F E G VG F E VG
C6 G VG E G VG F F VG F G E G
C7 P E VG G VG G G G VG F VG G
W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12
C1 VG E VG G E VG E E G G E G
C2 VG E E G E F VG G VG G E F
C3 P G G E G F G VG VG G E VG
C4 F G VG VG G G G F E VG VG VG
C5 G E E VG G F E VG G F E VG
C6 G VG E G G F F VG G F VG VG
C7 P E VG VG G VG VG G G G VG G

functional requirements, as a bare minimum for functional requirements. Table 5, Figure 4
and Figure 5 describe the linguistic terms, their abbreviations and fuzzy membership
functions.
As given in Table 5, in the evaluation process, 5-level scale was employed to translate
linguistic terms into fuzzy numbers. The judgments of the experts on alternatives are
illustrated in Tables 6,7 and 8.
The functional requirements of e-learning web sites are defined by three experts and
illustrated in Table 9.
Step 5. Fuzzification and aggregation of DMs’ opinions
DMs’ judgments on functional requirements and alternatives given in Step 4 are first
translated into fuzzy numbers and then aggregated using the methodology described in
Section 3.3. Table 10 displays aggregated evaluations on functional requirements and
alternatives.
Step 6. Computation of common area and information contents
Once fuzzified evaluations of DMs’ judgments are aggregated, fuzzy AD methodology
described in section 3.1 is applied to compute the common areas, which are shown in Table
11. The greater the common area, better is the response of the alternative to the functional
requirements.
Table 9. E-Learning web site functional requirements.
C1 C2 C3 C4 C5 C6 C7
DM1 LG LG LP LG LP LP LF
DM2 LG LVG LP LF LP LF LF
DM3 LG LVG LF LG LP LF LG

Computed common areas result in information contents. Table 12 displays information
contents for each alternative in response to each criteria and total information content for
each alternative. Alternatives that cannot meet functional requirements are eliminated as
there are no information content.
Table 10. Aggregated evaluations of DMs.
C1 C2 C3 C4 C5 C6 C7
FR 0.40 1.00 1.00 0.53 1.00 1.00 0.03 1.00 1.00 0.30 1.00 1.00 0.00 1.00 1.00 0.07 1.00 1.00 0.20 1.00 1.00
W1 0.53 0.68 0.83 0.67 0.83 0.93 0.06 0.11 0.36 0.14 0.24 0.44 0.26 0.41 0.56 0.34 0.49 0.64 0.06 0.11 0.36
W2 0.74 0.92 0.97 0.73 0.92 0.97 0.40 0.55 0.70 0.54 0.69 0.84 0.67 0.83 0.93 0.67 0.83 0.93 0.74 0.92 0.97
W3 0.60 0.75 0.90 0.73 0.91 0.96 0.47 0.62 0.77 0.74 0.92 0.97 0.74 0.92 0.97 0.74 0.92 0.97 0.54 0.69 0.84
W4 0.34 0.49 0.64 0.27 0.42 0.57 0.73 0.92 0.97 0.53 0.68 0.83 0.53 0.68 0.83 0.40 0.55 0.70 0.46 0.61 0.76
W5 0.74 0.92 0.97 0.73 0.92 0.97 0.40 0.55 0.70 0.53 0.68 0.83 0.27 0.42 0.57 0.46 0.61 0.76 0.54 0.69 0.84
W6 0.67 0.83 0.93 0.20 0.35 0.50 0.20 0.35 0.50 0.40 0.55 0.70 0.20 0.35 0.50 0.26 0.41 0.56 0.54 0.69 0.84
W7 0.74 0.92 0.97 0.53 0.68 0.83 0.27 0.42 0.57 0.53 0.68 0.83 0.73 0.92 0.97 0.26 0.41 0.56 0.54 0.69 0.84
W8 0.74 0.92 0.97 0.53 0.68 0.83 0.53 0.68 0.83 0.20 0.35 0.50 0.53 0.68 0.83 0.54 0.69 0.84 0.40 0.55 0.70
W9 0.53 0.68 0.83 0.67 0.83 0.93 0.60 0.75 0.90 0.73 0.91 0.97 0.53 0.68 0.83 0.27 0.42 0.57 0.54 0.69 0.84
W10 0.34 0.49 0.64 0.40 0.55 0.70 0.33 0.48 0.63 0.54 0.69 0.84 0.20 0.35 0.50 0.27 0.42 0.57 0.26 0.41 0.56
W11 0.73 0.92 0.97 0.73 0.91 0.97 0.73 0.91 0.96 0.73 0.92 0.97 0.73 0.92 0.97 0.73 0.92 0.97 0.66 0.83 0.93
W12 0.33 0.48 0.63 0.20 0.35 0.50 0.60 0.75 0.90 0.53 0.68 0.83 0.53 0.68 0.83 0.46 0.61 0.76 0.40 0.55 0.70
Table 11. Common areas (Ac)
C1 C2 C3 C4 C5 C6 C7
W1 0.11 0.11 0.04 0.01 0.10 0.10 0.01
W2 0.11 0.11 0.12 0.12 0.13 0.13 0.11
W3 0.12 0.11 0.13 0.11 0.12 0.11 0.13
W4 0.04 0.00 0.12 0.12 0.13 0.11 0.11
W5 0.11 0.11 0.12 0.12 0.10 0.12 0.13
W6 0.12 0.00 0.08 0.09 0.09 0.09 0.13
W7 0.11 0.07 0.09 0.12 0.12 0.09 0.13
W8 0.11 0.07 0.13 0.02 0.13 0.13 0.10
W9 0.11 0.11 0.14 0.11 0.13 0.09 0.13
W10 0.04 0.02 0.11 0.12 0.09 0.09 0.07
W11 0.11 0.11 0.12 0.11 0.12 0.12 0.13
W12 0.04 0.00 0.14 0.12 0.13 0.12 0.10
Table 12. Information contents
IC1 IC2 IC3 IC4 IC5 IC6 IC7 ITOT
W1 0.51 0.26 1.74 3.83 0.62 0.53 3.51 11.01
W2 0.03 0.07 0.36 0.34 0.04 0.05 0.02 0.92
W3 0.30 0.08 0.24 0.02 0.01 0.01 0.25 0.91
W4 2.02 7.49 0.01 0.34 0.16 0.39 0.39 10.81
W5 0.03 0.07 0.36 0.35 0.61 0.28 0.25 1.95
W6 0.14 ∞ 0.89 0.81 0.82 0.74 0.24 ∞
W7 0.03 1.02 0.66 0.34 0.01 0.74 0.24 3.06
W8 0.03 1.04 0.17 2.63 0.16 0.18 0.57 4.78
W9 0.51 0.25 0.10 0.03 0.16 0.73 0.25 2.03
W10 2.01 2.73 0.49 0.34 0.82 0.74 1.15 8.28
W11 0.04 0.08 0.01 0.03 0.01 0.01 0.07 0.25
W12 2.03 ∞ 0.10 0.34 0.16 0.28 0.57 ∞

Step 7. Calculation of the weighted information contents
Weighted information contents, given in Table 13, are calculated as described in section
3.1.
Step 8. Ranking the e-learning website alternatives
The final ranking is also given in Table 13. Final results demonstrate that two web sites
(W6 and W12) are eliminated, meaning that they do not meet the necessary functional
requirements. The evaluation results point out that web site W11 web site has the best
performance overall, followed by web site W2.
Step 9. Comparison with Fuzzy TOPSIS
As described in Section 3.4, fuzzy TOPSIS is applied to the aggregated decision matrix in
order to compare and justify the outcome of fuzzy AD. Tables 14 and 15 demonstrate
distances from FPIRP and FNIRP and total distances.
E-Learning web sites were ranked in increasing order of performance index. As seen in
Table 16, the outcome of the fuzzy AD methodology is justified with fuzzy TOPSIS. W11 is
ranked as the best alternative with both methodologies. Alternatives W6 and W12 are
Table 13. Weighted information contents.
W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12
WI 1.45 0.11 0.13 2.93 0.21 ∞ 0.54 0.84 0.28 1.58 0.05 ∞
Ranking 8 2 3 10 4 Eliminated 6 7 5 9 1 Eliminated
Table 14. Distances from FPIRP.
d+
1 d+
2 d+
3 d+
4 d+
5 d+
6 d+
7
Tot
d+
1
W1 0.89 0.72 0.99 0.96 0.97 0.96 0.97 6.47
W2 0.86 0.70 0.95 0.90 0.94 0.93 0.86 6.16
W3 0.88 0.70 0.95 0.87 0.94 0.93 0.89 6.17
W4 0.92 0.86 0.93 0.90 0.95 0.95 0.91 6.42
W5 0.86 0.70 0.95 0.90 0.97 0.95 0.89 6.24
W6 0.87 0.88 0.97 0.92 0.97 0.97 0.89 6.48
W7 0.86 0.77 0.97 0.90 0.94 0.97 0.89 6.29
W8 0.86 0.77 0.94 0.95 0.95 0.94 0.91 6.33
W9 0.89 0.72 0.94 0.87 0.95 0.97 0.89 6.24
W10 0.92 0.81 0.96 0.90 0.97 0.97 0.94 6.48
W11 0.87 0.70 0.93 0.87 0.94 0.93 0.87 6.11
W12 0.93 0.88 0.94 0.90 0.95 0.95 0.91 6.46
Table 15. Distances from FNIRP
d-
1 d-
2 d-
3 d-
4 d-
5 d-
6 d
W1 0.11 0.28 0.02 0.04 0.03 0.04 0.03
W2 0.14 0.30 0.05 0.10 0.06 0.07 0.14
W3 0.12 0.30 0.05 0.13 0.06 0.07 0.11
W4 0.08 0.15 0.07 0.10 0.05 0.05 0.10
W5 0.14 0.30 0.05 0.10 0.03 0.05 0.11
W6 0.13 0.13 0.03 0.08 0.03 0.04 0.11
W7 0.14 0.24 0.04 0.10 0.06 0.04 0.11
W8 0.14 0.24 0.06 0.05 0.05 0.06 0.09
W9 0.11 0.28 0.06 0.13 0.05 0.04 0.11
W10 0.08 0.19 0.04 0.10 0.03 0.04 0.07
W11 0.14 0.30 0.07 0.13 0.06 0.07 0.13
W12 0.08 0.13 0.06 0.10 0.05 0.05 0.09
Table 16. Performance indices for alternatives and ranking of the alternatives.
W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12

eliminated with fuzzy AD methodology given that these two alternatives cannot meet FR for
C2, complete content. However, as fuzzy TOPSIS evaluate the alternatives with respect to
FPIRP and FNIRP instead of a set of requirements determined by DMs, alternatives W6
and W12 are considered as well. This comparison of the outcome proves fuzzy AD to be a
more suitable methodology to evaluate a large number of alternatives, since a bare
minimum for the alternatives can be defined and unsuitable alternatives can easily be
eliminated.
5. Concluding remarks
Current developments of information systems facilitate greatly the diffusion of
knowledge. Knowledge and education form the source of more than 50% of the personal
national incomes of especially developed western countries and a well educated manpower
working at the jobs related with information (Kızılsu, 2006). On the other hand, the
advances in information systems and internet change the nature of education (Poon et al.,
2004). Therefore, education activities transfer to electronic platforms for higher speed and
less effort. Consequently, the web sites hosting the e-learning system become an important
interface for the end-user and web site quality directly affects e-learning provider’s
performance. Therefore, a performance analysis based on MDCM techniques is applied to
measure the quality of e-learning web sites. More precisely, in this paper, a group decision
based fuzzy AD methodology was applied to the problem of ranking e-learning web sites.
The proposed methodology is expected to provide additional contribution and decision
support to the managers working in the learning and e-business industries, due to its
advantages over already established techniques, such as fuzzy TOPSIS. Specifically, the
proposed methodology incorporates functional requirements into the ranking and selection
process, and can identify the alternatives that do not comply with the requirements.
For future research, the set of alternatives can be further extended and a two-stage
MCDM analysis consisting of pre-assessment and detailed evaluation can be applied in
order to thoroughly review e-learning web site alternatives. Given that service web sites are
increasing in number, pre-assessment stage will eliminate rapidly the unsuitable candidates
with general criteria and minimum assessment of alternatives, preferably with a single

expert. A more meticulous evaluation with fewer alternatives will be realized with a
hierarchic structure of criteria and a more detailed assessment of a group of expert. Also,
criteria set may be altered to evaluate web sites other than e-learning, such as e-commerce,
hospital, and bank web sites, based on the proposed methodology, since the general scheme
of criteria can be applied to all web sites with only a few changes.
Acknowledgements
The authors acknowledge the contribution of the expert committee members without which
this study could not be accomplished. Gülçin Büyüközkan acknowledges also the financial
support of the Galatasaray University Research Fund.
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